University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Mechanics (Physics 212A) Fall 2015
Assignment 7
Due 12:30pm Wednesday, November 18, 2015
1. Shape-invariant potentials. [Commins]
A particle of unit mass moves in a certain one-dimensional potential V− (x). The
boundstates u0 , u1 ...un have energies E0 , E1 ...En respectively, in order of increasing
energy. The groundstate wavefunction is
u0 ∝ (sechβx)p
where β, p > 0 are real parameters.
(a) Find the superpotential W (x).
(b) Find V− and its partner potential V+ .
(c) Obtain a general formula for the En . What restriction applies to p so that there
exist n boundstates?
(d) Find u1 , up to normalization.
(e) Use the result of part 1c to discuss the potential V = −V0 sech2 βx where V0 > 0,
and show the boundstate eigenvalues for m = ~ = 1 are given by
En = −
2
p
β2 −(1 + 2n) + 1 + 8V0 /β 2 .
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2. WKB vs exact solution.
Consider again the problem of a particle in a linear confining potential V (x) = |x|.
Compare the numerical values of the first three energy levels to the WKB approximation answers.
Consider the case of a quantum ball bouncing on the floor. Approximate the floor as
a hard wall.
How can you obtain the eigenfunctions for this case from the previous case?
Draw the phase space orbits for this case.
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3. Whittaker functions! [optional]
In deriving the connection formulae which relate WKB solutions to the Schrödinger
equation in neighboring allowed and forbidden regions, we assumed that the potential
had nonzero slope F0 at the turning point. What should you do if this assumption
fails, and the potential looks like this:
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